Math 257/316, Midterm 2, Section 101/102
November 20, 2009
Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.
Maximum score 100.
1. Solve the following inhomogeneous initial boundary value problem for the heat equation:
ut = uxx − u, 0 < x < 1, t > 0
ux (0, t) = 0, ux (1, t) = sinh(1)
u(x, 0) = cosh(x) − x
Hint: You might find it helpful to know that
Z
0
1
(
2/n2 π 2 , n odd
−x cos(nπx) dx =
0,
n even.
[50 marks]
2. Solve the following inhomogeneous initial boundary value problem for the heat equation:
ut
u(0, t)
u(x, 0)
= α2 uxx , 0 < x < π/2, t > 0
= 0, ux (π/2, t) = t2 /2
= 0
by using an appropriate expansion in terms of the eigenfunctions sin(2n+1)x corresponding to the eigenvalues
λn = (2n + 1) where n = 0, 1, . . .
Hint: You might find it helpful to know that
Z
π/2
−x sin ((2n + 1)x) dx = (−1)n+1 /(2n + 1)2
0
and that
Z
teγt dt = teγt /γ − eγt /γ 2
[50 marks]